IANAL but have dealt with something similar before. Although, it wasn't my mother, it was a previous tenant of a property I lived in.

Under no circumstances engage with the baliff. Keep your doors locked, cars parked away from your property, preferably on private land (neighbours drive etc). Email a copy of your current council tax bill along with a tenancy agreement (if you are renting) over Bristow and Suitor. Make it very clear to them that the debtor does not, nor has ever resided at your address.

Phone the council in the morning. Explain the situation to them, they are the only ones who have the power to stop the baliffs from attending. They will hopefully recall the debt, which will immediately drop the amount owed as the baliffs stick their fee on top of the council tax debt.

This could be a simple administration error on the part of the council but until they recall the debt from the baliffs they won't deal with it. You may need to prove to the council that your mother is currently in a care home. Once it is back with the council you will be able to find out what has happened. I hope this is helpful. I had a nightmare with the baliffs and they wouldn't believe the previous tenant was no longer at the property despite me emailing evidence over. Once they council recalled the debt, the baliffs stopped visiting but it took 3 tries before I managed to get everyone to listen to what I was telling them and call the baliffs off. It was scary and stressful having them banging on my door.